Currents of spin-polarized electrons can induce large-amplitude magnetization precessions at microwave frequencies in small-enough magnetic devices [Slonczewski 1996, Berger 1996]. There is mounting experimental evidence that these so-called spin-transfer phenomena do occur in nanopillar or nanocontact devices under current densities of the order of 10^{6}−10^{8} A/cm^{2} [Kiselev 2003, Rippard 2004, Krivorotov 2007, Boone 2009]. This discovery has boosted the already widespread interest in the physics of the interplay between magnetism and electron transport, and has triggered efforts toward the promising development of new generations of microwave spin-transfer nanooscillators.

A spin-transfer device is a nonlinear open system, driven far-from-equilibrium by the action of the spin-polarized electric current. The excited magnetization precessions represent strong excitations of the magnetic medium, which, in principle, may give rise to various types of instability and eventually to transitions to chaotic dynamics [Ji 2003, Polianski 2004, Zhu 2004, Lee 2004, Slavin 2008]. A parallel can be drawn with ferromagnetic-resonance Suhl’s instabilities [Suhl 1957] in which certain spin waves can get coupled to the uniform precession and start to grow to large nonthermal amplitudes, thus destroying the spatial uniformity of the original state.

In this letter, we demonstrate that spin-wave instabilities may occur in spin-transfer-driven magnetization dynamics as well. However, the system is far from equilibrium and the classical notion of spin waves fails. Indeed, it is the large-amplitude magnetization precession induced by spin transfer that plays the role of reference state, and spin waves only exist in a generalized, nonequilibrium sense, as small-amplitude perturbations of this state [Bertotti 2001, Kashuba 2006, Garanin 2009]. This scenario emerges with clarity in the time-dependent vector basis in which the reference magnetization precession is stationary. The spin-wave equations in this basis are characterized by two features: 1) a well-defined dispersion relation ω(*q*; cos θ_{0}), whose nonequilibrium nature is revealed by its explicit dependence on the magnetization precession amplitude cosθ_{0}; and 2) the presence of time-periodic coupling terms due to the magnetostatic fields generated by individual spin waves. This coupling leads to the appearance of narrow instability tongues around the parametric resonance condition ω (*q*; cos θ_{0}) ∼ ω_{0}, where ω_{0} is the magnetization precession angular frequency.

A preliminary, mostly mathematical analysis of the problem was proposed in [Bertotti 2009]. In this letter, we show that spin-wave instabilities occur only for particular combinations of external magnetic field and injected spin-polarized current. In addition, instabilities result in limited spatial and temporal distortions, which somewhat obscure, but yet do not completely disrupt the precessional character of the original state. This robustness of excited precessions with respect to spin-wave instabilities has a precise physical origin. Indeed, the discrete nature of the spin-wave spectrum caused by boundary conditions in submicrometer devices reduces the number of available spin-wave modes, which can contribute to instabilities. On the other hand, the strength of the magnetostatic effects responsible for instabilities is drastically reduced, due to the ultrathin nature of spin-transfer devices, and instability thresholds are consequently enhanced. Finally, spin-transfer-driven precessions are characterized by large amplitudes, and as such, are less easily masked by the onset of nonuniform modes. In spin-transfer nanooscillators, spin-wave instabilities are expected to result in increased oscillator linewidths, a conclusion that might explain some of the puzzling experimental results obtained in this area [Mistral 2006].

SECTION II

## PHYSICAL MODEL

To start the technical discussion, consider an ultrathin disk with negligible crystal anisotropy (e.g., permalloy). Typically, this disk will be the so-called free layer of a nanopillar spin-transfer device (see inset in Fig. 1). The disk plane is parallel to the (*x*,*y*) plane and is traversed by a flow of electrons with spin polarization along the **e**_{z} direction. The dimensionless equation for the dynamics of the normalized magnetization **m**(**r**,*t*) (|**m**|^{2} = 1) in the disk, in the presence of spin transfer and neglecting the Oersted field due to the electric current, is as follows [Slonczewski 1996, Bertotti 2005]:
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$$\eqalignno{{\partial {\bf m} \over \partial t} &- \alpha \, {\bf m}\times { \partial {\bf m}\over \partial t} = - {\bf m}\times ({\rm h}_{az}{\bf e}_{z}+ {\bf h}_{M}+ \nabla^2 {\bf m}- \beta \, {\bf m}\times {\bf e}_{z}).\qquad&\hbox{(1)}}$$Here, the external magnetic field h_{az}**e**_{z} and the magnetostatic field **h**_{M} are measured in units of the spontaneous magnetization M_{s}, time in units of (γ M_{s})^{−1} (γ is the absolute value of the gyromagnetic ratio), and lengths in units of the exchange length. The field **h**_{M} is the solution to magnetostatic Maxwell equations ∇ × **h**_{M} = **0**, ∇ ·**h**_{M} = −∇ ·**m** with appropriate interface conditions. The external field is perpendicular to the disk plane, while the spin-transfer torque is simply proportional to the sine of the angle between **m** and **e**_{z}. The parameter β is proportional to the spin-polarized current density (see [Bertotti 2005] for the detailed definition), and in typical situations, it is comparable with the damping constant α.

Whenever |h_{az}− β/α | ≤ *N*_{z}− *N*_{⊥} (*N*_{z} and *N*_{⊥} are the disk demagnetizing factors, with *N*_{z}+ 2 *N*_{⊥} = 1), (1) admits time-harmonic solutions **m**_{0}(*t*), corresponding to spatially uniform precession of the magnetization around the *z*-axis [Bazaliy 2004] (see Fig. 1). The precession amplitude and angular frequency are as follows:
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$$\cos \theta_0 = { {\rm h}_{az}- \beta /\alpha \over N_{z}- N_{\bot }}, \omega_0 = { \beta \over \alpha }.\eqno{\hbox{(2)}}$$

To study the stability of **m**_{0}(*t*), consider the perturbed motion **m**(**r**, *t*) = **m**_{0}(*t*) + δ**m**(**r**, *t*), with | δ**m**(**r**, *t*) | ≪ 1. The corresponding magnetostatic field will be: **h**_{M}(**r**, *t*) = − *N*_{z}**m**_{0 z}− *N*_{⊥}**m**_{0 ⊥}+ δ**h**_{M}(**r**, *t*), where δ**h**_{M} represents the magnetostatic field generated by δ**m**. Since, we are interested in ultrathin layers, we shall assume that δ**m** does not depend on *z*: δ**m**(**r**,*t*) = δ**m**(*x*,*y*,*t*).

The perturbation δ**m** is orthogonal to **m**_{0}(*t*) at all times, since the local magnetization magnitude |**m**|^{2} = 1 must be preserved. Hence, it is natural to represent δ**m** in the time-dependent vector basis (**e**_{1}(*t*),**e**_{2}(*t*) ) defined in the plane perpendicular to **m**_{0}(*t*), with **e**_{2}(*t*) parallel to **e**_{z}×**m**_{0}(*t*) and **e**_{1}(*t*), such that (**e**_{1},**e**_{2},**m**_{0}) form a right-handed orthonormal basis. The perturbation can be written as: δ**m**(**r**, *t*) = δ *m*_{1}(**r**, *t*) **e**_{1}(*t*) + δ *m*_{2}(**r**, *t*) **e**_{2}(*t*). By linearizing (1) around **m**_{0}(*t*) and averaging the linearized equation over the layer thickness, one obtains the following coupled differential equations in matrix form:
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$$\eqalignno{\left(\!\matrix{1 & \alpha\cr - \alpha & 1}\!\right) {\partial \over \partial t} \left(\!\matrix{\delta m_{1}\cr \delta m_{2}}\!\right) &= \left(\!\matrix{0 & 1\cr -1 & 0}\!\right) \left(\!\matrix{\langle \delta {\rm h}_{M}\rangle_1\cr\langle \delta {\rm h}_{M}\rangle_2} \!\right) \cr&\quad + \left(\!\matrix{0 & N_{\bot }+ \nabla^{2}_{\bot }\cr - N_{\bot }- \nabla^{2}_{\bot} & 0}\!\right) \left(\!\matrix{\delta m_{1}\cr \delta m_{2}}\!\right)\qquad&\hbox{(3)}}$$where ∇^{2}_{⊥} = ∂^{2}/∂ *x*^{2} + ∂^{2}/∂ *y*^{2}, while < … > represents the *z* average over the thickness of the disk, and < δh_{M}>_{1} = < δ**h**_{M}> ·**e**_{1}(*t*), < δh_{M}>_{2} = < δ**h**_{M}> ·**e**_{2}(*t*).

To grasp the physical consequences of (3), consider the plane-wave perturbation δ**m**(**r**, *t*) = *a*(*t*) exp (*i* **q**·**r**) in an infinite layer (*N*_{⊥} = 0). The corresponding magnetostatic field is as follows [Bertotti 2009]:
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$$\langle \delta {\bf h}_{M}\rangle = - s_q \, \delta {\bf m}_z - \left(1 - s_{q} \right) \delta {\bf m}_q; s_{q} = { 1 - \exp (-q d)\over q d}\eqno{\hbox{(4)}}$$where δ**m**_{z} = (δ**m**·**e**_{z})**e**_{z} and δ**m**_{q} = (δ**m**·**e**_{q})**e**_{q}, **e**_{q} being the unit vector in the **q** direction. The field − *s*_{q} δ**m**_{z} is generated by the magnetic charges at the layer surface, whereas the field − (1 − *s*_{q}) δ**m**_{q} is due to volume charges. By taking into account that ∇_{⊥}^{2} δ**m** = − *q*^{2} δ**m**, δ**m**_{z} ·**e**_{1}(*t*) = δ *m*_{1} sin^{2} θ_{0}, and δ**m**_{z} ·**e**_{2}(*t*) = 0, one finds from (3) that **m**_{0}(*t*) is always stable with respect to the action of exchange forces and surface magnetic charges. Only volume charges can make the precession unstable. This conclusion follows from the fact that the *z*-axis, along which the surface-charge magnetostatic field is directed, is a symmetry axis for the problem. Surface-charge-driven instabilities may appear in nonuniaxial systems.

The 2-D and uniaxial character of the problem makes it natural to introduce polar coordinates (*r*, ϕ) in the disk plane, with the origin at the center of the disk. The natural boundary condition in polar coordinates is ∂δ**m**/∂ *r* |_{r = R} = 0, where *R* is the disk radius. The generic perturbation satisfying this boundary condition consists of cylindrical spin waves of the type
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$$\delta {\bf m}(r, \phi, t) = \sum_{n = -\infty }^{+\infty } \sum_{k = 0}^{\infty } {\bf {a}}_{n k} (t) \, J_n \left(q_{n k} r \right) \, \exp (i n \phi)\eqno{\hbox{(5)}}$$where *J*_{n} (*z*) is the *n*th order Bessel function. The wave-vector amplitude *q*_{nk} is identified by two subscripts because, for each *n*, it must satisfy the boundary condition ∂J_{n} (*z*)/∂ *z* = 0 for *z* = *q*_{nk} *R*, which has infinite solutions *q*_{n 0}, *q*_{n1}, *q*_{n2}, … of increasing amplitude. The cylindrical spin waves *F*_{nk}(*r*, ϕ) = *J*_{n} (*q*_{nk} *r* ) exp (*in* ϕ) are a complete orthogonal set of eigenfunctions of the ∇^{2}_{⊥} operator: ∇^{2}_{⊥} *F*_{nk}(*r*, ϕ) = − *q*_{nk}^{2} *F*_{nk}(*r*, ϕ). The magnetostatic field δ**h**_{M} can be computed by applying (4) to the plane wave integral representation: *F*_{nk}(*r*, ϕ) = 1/(2 π *i*^{n}) ∫_{0}^{2 π}exp (*i* **q**_{nk}·**r**) exp (*in* ψ) *d* ψ, where the polar representation of **r** and **q**_{nk} is **r** = (*r*, ϕ) and **q**_{nk} = (*q*_{nk}, ψ), respectively. By following these steps, writing *a*_{nk}(*t*) as *a*_{nk}(*t*) = *c*_{nk,1}(*t*) **e**_{1}(*t*) + *c*_{nk,2}(*t*) **e**_{2}(*t*), and neglecting small terms proportional to *N*_{⊥}, (3) is transformed into the following system of coupled equations:
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$$\eqalignno{{ d c_{n k}\over d t} &= A_{n k} \, c_{n k} + \exp \left(2 i \omega_0 t \right) \,{\cal R} \,\sum_{p = 0}^{\infty } \, \Delta_{n k; p}^{+} \, \epsilon_{n+2,p} \, c_{n+2,p}\cr&\quad + \exp \left(- 2 i \omega_0 t \right) \,{\cal R}^* \,\sum_{p = 0}^{\infty } \, \Delta_{n k; p}^{-} \, \epsilon_{n-2,p} \, c_{n-2,p}&\hbox{(6)}}$$where
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$$\eqalignno{c_{n k} &\equiv \left(\matrix{ c_{nk,1}\cr c_{nk,2}}\right)&{\hbox{(7)}}\cr A_{n k} &= { 1\over 1 + \alpha^{2}} \left(\matrix{ 1 & - \alpha\cr \alpha & 1}\right) \left(\matrix{ 0 & - \nu_{n k}\cr \nu_{n k} - \kappa_{n k} \sin^{2} \theta_{0} & 0}\right)&{\hbox{(8)}}\cr{\cal R} &= { 1\over 1 + \alpha^{2}} \, \left(\matrix{ 1 & - \alpha\cr \alpha & 1}\right) \left(\matrix{ i \, \cos \theta_{0} & -1\cr - \cos^{2} \theta_{0} & - i \, \cos \theta_{0}}\right)&{\hbox{(9)}}\cr\Delta_{n k; p}^{\pm } &={ 1\over \Delta_{n k}} \int_{0}^{R} r \, J_{n}\left(q_{n k} r \right) \, J_{n}\left(q_{n \pm 2, p} r \right) \, d r&{\hbox{(10)}}}$$and Δ_{nk} = ∫_{0}^{R} *r* *J*_{n}^{2} (*q*_{nk} *r*) *dr*, ν_{nk} = *q*_{nk}^{2}+ 2 ∊_{nk}, κ_{nk} = − 1 + 6 ∊_{nk}, ∊_{nk} = (1−*s*_{nk})/4, *s*_{nk} being the value of *s*_{q} in (4) for *q* = *q*_{nk}.

SECTION III

## RESULTS AND DISCUSSION

The coupling terms proportional to in (6) are the consequence of volume–charge magnetostatic effects. They are all of the order of (1 − *s*_{q} ). One has that (1 − *s*_{q}) ≪ 1 up to *q* ∼ 1 in ultrathin layers with [see (4)]. If one neglects these terms altogether, one obtains a system of fully decoupled equations for individual cylindrical spin waves, characterized by the dispersion relation
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$$\eqalignno{\omega^2 (q; \cos \theta_0) &= \left(\!q^2 + { 1 - s_q\over 2} \!\right)\! \left(\!q^2 + s_q + { 1 - 3 s_q\over 2} \, \cos^2 \theta_0 \!\right)\qquad&\hbox{(11)}}$$which is obtained from (8) in the limit α → 0. However, the time-periodic coupling terms may give rise to parametric instabilities. Interestingly, these instabilities are governed by a small number of dominant terms, which can be identified by using the asymptotic formula in the equation expressing boundary conditions. One obtains the estimate *q*_{nk}≃ π (2 *s* + 1)/4 *R*, where *s* = | *n*| + 2 *k*. When this approximate expression is used for *q*_{n ± 2, p} in (10), one obtains
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$$\eqalignno{n \ge 2 & :\quad \Delta_{n k; p}^{\pm } \simeq \delta_{p, k \mp 1},\quad \Delta_{n 0; p}^{+} \simeq 0,\cr n = \pm 1 & :\quad \Delta_{1k; p}^{-} = \Delta_{-1,k; p}^{+} = \delta_{p k},\cr n \le -2 & :\quad \Delta_{n k; p}^{\pm } \simeq \delta_{p, k \pm 1}, \quad \Delta_{n 0; p}^{-} \simeq 0.&\hbox{(12)}}$$

These relations have an important physical consequence, which is best appreciated by rewriting (5) in the form: δ**m** = ∑_{s = 0}^{∞}δ**m**^{(s)}, where
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$$\delta {\bf m}^{(s)} = \displaystyle \sum_{\vert n\vert + 2k = s} {\bf {a}}_{n k} (t) \, J_n \left(q_{n k} r \right) \, \exp (i n \phi).\eqno{\hbox{(13)}}$$Under the approximation (12), one finds from (6) that δ**m**^{(s1)} is decoupled from δ**m**^{(s2)} for any *s*_{2} ≠ *s*_{1}. On the other hand, for each *s*, the (*s*+1) cylindrical waves (namely, *n* = *s*, *s*−2, …, −*s* +2, −*s*) involved in δ**m**^{(s)} form a 1-D chain, in the sense that only neighboring waves in the aforementioned list are coupled. The absence of coupling between distinct chains would be complete if the approximation *q*_{nk}≃ π (2 *s* + 1)/4 *R* were exact. In this case, all the cylindrical waves in δ**m**^{(s)} would be characterized by exactly the same wave-vector amplitude.

Instabilities are governed by the multipliers of the one-period map [Perko 1996] associated with the dynamics of δ**m**^{(s)}. We have used (6) and (12) to make a numerical study of these multipliers for different chains, in order to obtain the instability pattern associated with each of them. The results for a permalloy disk with radius *R* = 135 nm and thickness *d* = 3.43 nm are shown in Fig. 1. The band (**O**+**SW**) in between the **P** and **A** regions is where magnetization precession occurs. Spin-wave instabilities appear in region **SW**. Each chain δ**m**^{(s)} provides a distinct instability channel. In particular, chains *s* = 1, *s* = 2, and *s* = 3 (*s* = 0 yields no instability at all) give rise to well-separated instability regions that can be neatly resolved, as shown in Fig. 2(a). In general, the *s*th chain gives rise to an instability tongue around the parametric resonance condition ω (*q*; cos θ_{0}) ∼ ω_{0}, where ω (*q*; cos θ_{0}) is given by (11) and *q* is of the order of the wave-vector amplitudes involved in the chain [see dashed lines in Fig. 2(a)]. According to parametric resonance theory, resonance occurs for ω = *n* ω_{0}/2, *n* = 1, 2, …. The dominant, lowest threshold resonance occurs for *n* = 1, i.e., at ω = ω_{0}/2. However, one can see from (6) that the parametric frequency is 2 ω_{0} rather than ω_{0}, which explains why the resonance condition is ω ∼ ω_{0}. This peculiarity is the consequence of the rotational invariance of the problem, and is expected to disappear in situations with broken rotational symmetry. Interestingly, Fig. 1 reveals that, for a given precession amplitude cos θ_{0} (see dashed line), applying larger fields and currents has a stabilizing effect on the precession. Also, larger fields stabilize precessions of given frequency ω_{0} = β/α.

To test the predictions of the theory, we have carried out computer simulations based on the numerical integration of (1) by the methods discussed in [d'Aquino 2005]. Simulations were carried out by slowly varying the external magnetic field under constant current. As shown in Fig. 2(b), at large fields the magnitude *m*_{⊥}^{avg} of the average in-plane magnetization is in full agreement with the prediction of (2) for spatially uniform precession. Then, under decreasing field *m*_{⊥}^{avg} exhibits well-pronounced jumps, whose positions agree with the theoretical instability thresholds for the *s* = 2 and *s* = 3 chains within ten percentage. Beyond these jumps, nonuniform modes appear in the dynamics [see Fig. 2(b)], characterized by twofold and threefold patterns that are consistent with the symmetry of the cylindrical waves involved in the *s* = 2 and *s* = 3 chains, respectively. It is worth remarking that the main precession is not completely disrupted, but only somewhat obscured by spin-wave instabilities. Agreement with the theory is also confirmed by the hysteresis in the instability thresholds occurring under decreasing or increasing external field [see Fig. 2(c)]. Future work will be devoted to extending the present approach to more general, nonuniaxial geometries.

### Acknowledgment

This work was supported in part by the Ministry of Scientific Research through the Project PRIN-2006098315 (Italy), in part by the European Union (European Social Fund), and in part by the Regione Autonoma Valle d’Aosta (Italy).